10.1 Molecular Diffusion, Fick's Laws, and Equimolar Counterdiffusion

10.1 Molecular Diffusion, Fick's Laws, and Equimolar Counterdiffusion

On the NCEES PE Chemical Exam, mass transfer fundamentals provide the analytical bedrock for rating and sizing distillation columns, gas absorbers, liquid-liquid extractors, and membrane separators. While fluid mechanics dictates the macroscopic convective flow of process streams, separation at the phase boundary or within stagnant boundary layers is ultimately constrained by molecular diffusion—the spontaneous transport of individual chemical species down a gradient of chemical potential or concentration resulting from random thermal molecular motion.

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1. Physical Foundations of Diffusion and Reference Frames

To analyze mass transport rigorously, chemical engineers distinguish between two coordinate reference frames:

1. A reference frame moving with the molar-average velocity of the fluid mixture ($v^*$). In this moving frame, mass transport occurs purely by molecular diffusion.
2. A fixed laboratory coordinate reference frame (stationary observer). In this frame, mass transport represents the vector sum of bulk convective flow (drift) plus molecular diffusion.

+---------------------------------------------------------------------------------+
|                      TOTAL MOLAR FLUX IN FIXED COORDINATES                      |
|                                                                                 |
|        N_A         =          x_A * (N_A + N_B)         -   C * D_AB * dx_A/dz  |
|         ^                            ^                               ^          |
|   [Total Molar Flux]       [Convective Bulk Drift]          [Molecular Diffusion]|
|  (relative to fixed     (carried by net bulk motion        (Fick's First Law relative|
|     coordinates)          of all diffusing species)         to molar-average velocity)|
+---------------------------------------------------------------------------------+

Molar-Average Velocity and Diffusive Flux ($J_A^*$)

For a binary mixture composed of species $A$ and $B$, let $u_A$ and $u_B$ be the absolute linear velocities of the respective species in the $z$-direction relative to a fixed stationary boundary. The absolute molar fluxes relative to stationary coordinates are:

$$N_A = C_A u_A, \quad N_B = C_B u_B$$

Where:

• $N_A, N_B$ = molar fluxes relative to fixed coordinates ($\text{kmol/(m}^2\cdot\text{s)}$ or $\text{lbmol/(ft}^2\cdot\text{hr)}$).
• $C_A, C_B$ = molar concentrations of species $A$ and $B$ ($\text{kmol/m}^3$ or $\text{lbmol/ft}^3$).
• $C = C_A + C_B$ = total molar concentration of the mixture.
• $x_A = C_A / C, \quad x_B = C_B / C$ = mole fractions ($x_A + x_B = 1.0$).

The mixture molar-average velocity ($v^*$) is defined as:

$$v^* = \frac{C_A u_A + C_B u_B}{C} = x_A u_A + x_B u_B = \frac{N_A + N_B}{C}$$

The diffusive molar flux of species $A$ relative to the moving molar-average velocity frame is denoted $J_A^*$ and is governed by Fick's First Law of Diffusion:

$$J_A^* = C_A (u_A - v^*) = -C D_{AB} \frac{dx_A}{dz}$$

Where:

• $D_{AB}$ = binary molecular diffusivity of species $A$ diffusing through species $B$ ($\text{m}^2/\text{s}$ or $\text{ft}^2/\text{hr}$).
• In a binary system, thermodynamics and continuity dictate that $D_{AB} = D_{BA}$ and $J_A^* + J_B^* = 0$.

The General 1-D Absolute Molar Flux Equation

Substituting $v^* = (N_A + N_B)/C$ into the definition of $J_A^*$ yields:

$$N_A = C_A v^* + J_A^* = x_A (N_A + N_B) - C D_{AB} \frac{dx_A}{dz}$$

For an ideal gas phase, where $C = P / (R T)$, $x_A = y_A = p_A / P$, and $C_A = p_A / (R T)$:

$$N_A = y_A (N_A + N_B) - \frac{D_{AB}}{R T} \frac{dp_A}{dz}$$

Where:

• $P$ = total system absolute pressure ($\text{kPa}$, $\text{bar}$, or $\text{atm}$).
• $p_A$ = partial pressure of species $A$ ($\text{kPa}$, $\text{bar}$, or $\text{atm}$).
• $R$ = universal gas constant ($8.31446\text{ J/(mol}\cdot\text{K)} = 8.31446\text{ kPa}\cdot\text{m}^3/(\text{kmol}\cdot\text{K)} = 0.082057\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K)}$).
• $T$ = absolute thermodynamic temperature ($\text{K}$ or $^\circ\text{R}$).

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2. Equimolar Counterdiffusion (EMCD)

Defining Characteristics

Equimolar Counterdiffusion (EMCD) occurs when one mole of species $A$ diffuses in the positive $z$-direction while exactly one mole of species $B$ diffuses in the opposite direction:

$$N_B = -N_A \implies N_A + N_B = 0$$

Industrial and Physical Examples

1. Binary Distillation of Similar Compounds: When latent heats of vaporization are approximately equal ($\Delta H_{vap,A} \approx \Delta H_{vap,B}$), the condensing vapor of the less volatile component releases precisely enough latent heat to vaporize an equimolar quantity of the more volatile liquid component from the tray.
2. Diffusion Between Two Connected Gas Vessels: Two large vessels maintained at identical total pressure and temperature containing different gases connected by a narrow capillary tube.

Derivation of EMCD Flux and Concentration Profile

Because $N_A + N_B = 0$, the bulk convective drift term collapses to zero:

$$N_A = -C D_{AB} \frac{dx_A}{dz} = -\frac{D_{AB}}{R T} \frac{dp_A}{dz}$$

At steady state with no chemical reactions, species conservation requires that $d N_A / dz = 0$. For an ideal gas at constant temperature and pressure, $C$ and $D_{AB}$ are constant. Separating variables and integrating across a stagnant boundary layer of thickness $z = z_2 - z_1 = \delta$:

$$\int_{z_1}^{z_2} N_A dz = -\frac{D_{AB}}{R T} \int_{p_{A1}}^{p_{A2}} dp_A$$ $$N_A \cdot (z_2 - z_1) = -\frac{D_{AB}}{R T} (p_{A2} - p_{A1}) = \frac{D_{AB}}{R T} (p_{A1} - p_{A2})$$ $$\mathbf{N_A = \frac{D_{AB}}{R T z} (p_{A1} - p_{A2}) = \frac{C D_{AB}}{z} (y_{A1} - y_{A2})}$$

Integrating from $z = 0$ to an arbitrary distance $z$ demonstrates that the partial pressure profile under EMCD is strictly linear:

$$p_A(z) = p_{A1} - \left( \frac{p_{A1} - p_{A2}}{z_2 - z_1} \right) z$$

   Partial Pressure (p_A) ^
                      p_A1 |* 
                           |  * 
                           |    *  <--- Strictly Linear Profile (EMCD)
                           |      * 
                      p_A2 |________*_________
                           0                 z (Distance)

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3. Diffusion of A Through Stagnant, Non-Diffusing B ($N_B = 0$)

Defining Characteristics

In many separation units, one component moves while the surrounding carrier medium remains stationary:

$$N_B = 0 \implies N_A + N_B = N_A$$

Industrial and Physical Examples

1. Stefan Tube Evaporation: Liquid volatile solvent evaporating from a capillary or deep pool into an inert, insoluble gas blanket (e.g., liquid benzene or water evaporating into dry air).
2. Gas Absorption / Scrubbing: Absorption of a highly soluble gas (e.g., ammonia or hydrogen chloride) from an insoluble carrier gas (air or methane) into an aqueous wash stream. The carrier gas does not dissolve ($N_B = 0$).
3. Catalytic Surface Reactions with Non-Equimolar Stoichiometry: E.g., $2A \to B$ or a cracking reaction where species molar fluxes differ.

Derivation of Stagnant-B Flux and the Stefan Drift

Setting $N_B = 0$ in the general flux equation:

$$N_A = y_A N_A - \frac{D_{AB}}{R T} \frac{dp_A}{dz}$$

Rearranging to isolate $N_A$:

$$N_A (1 - y_A) = -\frac{D_{AB}}{R T} \frac{dp_A}{dz}$$

Recognizing that in a binary gas mixture $y_A = p_A / P$, the factor $(1 - y_A) = (P - p_A)/P = p_B / P$:

$$N_A \left( \frac{p_B}{P} \right) = -\frac{D_{AB}}{R T} \frac{dp_A}{dz}$$

Because total pressure is uniform ($P = p_A + p_B = \text{constant}$), differentiating gives $dp_A = -dp_B$:

$$N_A \left( \frac{p_B}{P} \right) = \frac{D_{AB}}{R T} \frac{dp_B}{dz}$$

Separating variables across the diffusion path length from $z = 0$ (where $p_A = p_{A1}, p_B = p_{B1}$) to $z = \delta$ (where $p_A = p_{A2}, p_B = p_{B2}$):

$$\int_0^\delta N_A dz = \frac{D_{AB} P}{R T} \int_{p_{B1}}^{p_{B2}} \frac{dp_B}{p_B}$$ $$N_A \delta = \frac{D_{AB} P}{R T} \ln\left( \frac{p_{B2}}{p_{B1}} \right) = \frac{D_{AB} P}{R T} \ln\left( \frac{P - p_{A2}}{P - p_{A1}} \right)$$ $$\mathbf{N_A = \frac{D_{AB} P}{R T z} \ln\left( \frac{P - p_{A2}}{P - p_{A1}} \right)}$$

The Log-Mean Partial Pressure of the Stagnant Gas ($p_{B,lm}$)

To cast the stagnant diffusion equation into a form that directly parallels EMCD, we introduce the log-mean partial pressure of the non-diffusing component $B$:

$$p_{B,lm} = \frac{p_{B2} - p_{B1}}{\ln(p_{B2} / p_{B1})}$$

Since $p_{B2} - p_{B1} = (P - p_{A2}) - (P - p_{A1}) = p_{A1} - p_{A2}$, solving for the logarithmic term gives:

$$\ln\left( \frac{p_{B2}}{p_{B1}} \right) = \frac{p_{A1} - p_{A2}}{p_{B,lm}}$$

Substituting this identity back into the flux equation yields the classical engineering expression:

$$\mathbf{N_A = \frac{D_{AB} P}{R T z p_{B,lm}} (p_{A1} - p_{A2}) = \frac{C D_{AB}}{z} \left( \frac{P}{p_{B,lm}} \right) (y_{A1} - y_{A2})}$$

The Drift Enhancement Multiplier ($P / p_{B,lm}$)

Comparing the two fundamental flux expressions for an identical partial pressure driving force $(p_{A1} - p_{A2})$:

$$\frac{N_{A,\text{stagnant}}}{N_{A,\text{EMCD}}} = \frac{P}{p_{B,lm}}$$

Because $p_{B,lm}$ represents the average partial pressure of component $B$, and since $p_B < P$, the ratio $P / p_{B,lm}$ is always strictly greater than $1.0$.

• For dilute systems where $y_A \ll 1$ (e.g., $y_A < 0.05$), $p_B \approx P$, so $P / p_{B,lm} \approx 1.0$, and the stagnant flux equation converges to the EMCD equation.
• For concentrated systems where $y_A$ is large, the Stefan bulk flow carries substantial solute toward the boundary, significantly amplifying the net flux.

Non-Linear Concentration Profile in Stagnant Media

Unlike EMCD, the concentration profile for diffusion through stagnant $B$ is non-linear (exponential) due to convective bulk drag:

$$\frac{P - p_A(z)}{P - p_{A1}} = \frac{p_B(z)}{p_{B1}} = \left( \frac{p_{B2}}{p_{B1}} \right)^{z / \delta}$$

   Partial Pressure (p_A) ^
                      p_A1 |* 
                           |  ` . 
                           |      ` .  <--- Concave-Up Profile (Stagnant B)
                           |          ` .
                      p_A2 |_____________*____
                           0                 z (Distance)

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4. Transient Molecular Diffusion & Fick's Second Law

When concentration changes with respect to both position and time, species conservation in a stationary medium with constant $D_{AB}$ is governed by Fick's Second Law:

$$\frac{\partial C_A}{\partial t} = D_{AB} \frac{\partial^2 C_A}{\partial z^2}$$

Semi-Infinite Medium Solution

Consider a semi-infinite liquid or solid medium initially at uniform concentration $C_{A0}$. At $t = 0$, the surface at $z = 0$ is suddenly exposed to a constant surface concentration $C_{As}$. The boundary and initial conditions are:

• $C_A(z, 0) = C_{A0}$ for $z \ge 0$
• $C_A(0, t) = C_{As}$ for $t > 0$
• $C_A(\infty, t) = C_{A0}$ for $t > 0$

The analytical solution is formulated using the complementary error function ($\text{erfc}$):

$$\frac{C_A(z, t) - C_{A0}}{C_{As} - C_{A0}} = \text{erfc}\left( \frac{z}{2 \sqrt{D_{AB} t}} \right) = 1 - \text{erf}\left( \frac{z}{2 \sqrt{D_{AB} t}} \right)$$

Instantaneous Surface Molar Flux

Evaluating the flux at the interface ($z = 0$) using Fick's first law:

$$N_A(0, t) = -D_{AB} \left. \frac{\partial C_A}{\partial z} \right|_{z=0} = \sqrt{\frac{D_{AB}}{\pi t}} (C_{As} - C_{A0})$$

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5. Estimation of Diffusivity in Gases, Liquids, and Porous Media

1. Fuller-Schettler-Giddings (FSG) Correlation for Binary Gases

For low-pressure binary gas mixtures ($P < 10\text{ bar}$), the semi-empirical Fuller-Schettler-Giddings (FSG) method provides superior accuracy (typical error $< 5\%$):

$$D_{AB} = \frac{1.013 \times 10^{-7} \cdot T^{1.75} \left( \frac{1}{M_A} + \frac{1}{M_B} \right)^{0.5}}{P \left[ \left(\sum v\right)_A^{1/3} + \left(\sum v\right)_B^{1/3} \right]^2}$$

Where:

• $D_{AB}$ = binary gas diffusivity ($\text{m}^2/\text{s}$).
• $T$ = absolute temperature ($\text{K}$).
• $P$ = absolute pressure ($\text{bar}$).
• $M_A, M_B$ = molecular weights ($\text{g/mol}$).
• $\sum v$ = sum of atomic diffusion volumes (e.g., $\text{C} = 15.9, \text{H} = 2.31, \text{O} = 6.11, \text{N} = 4.54, \text{Air} = 19.7$).

Scaling Relations for Gases:

$$D_{AB} \propto \frac{T^{1.75}}{P} \implies D_{AB}(T_2, P_2) = D_{AB}(T_1, P_1) \left( \frac{T_2}{T_1} \right)^{1.75} \left( \frac{P_1}{P_2} \right)$$

2. Wilke-Chang Correlation for Dilute Liquid Solutions

Liquid diffusion is constrained by molecular packing and solvent viscosity. The Wilke-Chang correlation applies to small solute molecules ($A$) diffusing through dilute liquid solvent ($B$):

$$D_{AB} = \frac{7.4 \times 10^{-8} \cdot (\phi M_B)^{0.5} \cdot T}{\mu_B \cdot V_A^{0.6}}$$

Where:

• $D_{AB}$ = liquid diffusivity ($\text{cm}^2/\text{s}$; multiply by $10^{-4}$ to convert to $\text{m}^2/\text{s}$).
• $\phi$ = solvent association factor:
  • Water: $\phi = 2.6$
  • Methanol: $\phi = 1.9$
  • Ethanol: $\phi = 1.5$
  • Unassociated solvents (benzene, hexane, toluene, diethyl ether): $\phi = 1.0$
• $M_B$ = molecular weight of solvent ($\text{g/mol}$).
• $T$ = absolute temperature ($\text{K}$).
• $\mu_B$ = dynamic viscosity of solvent ($\text{cP}$; $1\text{ cP} = 10^{-3}\text{ Pa}\cdot\text{s} = 1\text{ mPa}\cdot\text{s}$).
• $V_A$ = molar volume of solute at its normal boiling point ($\text{cm}^3/\text{mol}$).

Scaling Relations for Liquids: Because liquids are essentially incompressible, liquid diffusivity is practically independent of pressure at industrial pressures ($< 50\text{ bar}$). Temperature scaling depends heavily on solvent viscosity:

$$D_{AB} \propto \frac{T}{\mu_B(T)} \implies D_{AB}(T_2) = D_{AB}(T_1) \left( \frac{T_2}{T_1} \right) \left( \frac{\mu_B(T_1)}{\mu_B(T_2)} \right)$$

3. Knudsen Diffusion in Porous Solids

When gas diffusion occurs within micro-porous catalyst pellets or zeolites where pore diameter ($d_{pore}$) is smaller than the molecular mean free path ($\lambda$):

$$\lambda = \frac{k_B T}{\sqrt{2} \pi d_{mol}^2 P}$$

The Knudsen number is defined as $Kn = \lambda / d_{pore}$:

• $Kn < 0.01$: Fickian molecular bulk diffusion dominates; collisions occur between gas molecules.
• $0.01 \le Kn \le 10$: Transition regime; both pore wall collisions and intermolecular collisions matter.
• $Kn > 10$: Knudsen diffusion dominates; molecules collide almost exclusively with pore walls.

The Knudsen diffusivity ($D_{K,A}$) is given by kinetic theory:

$$D_{K,A} = \frac{d_{pore}}{3} \sqrt{\frac{8 R T}{\pi M_A}} = 48.5 \cdot d_{pore} \sqrt{\frac{T}{M_A}}$$

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6. Summary Comparison Table: Diffusion Regimes and Transport Equations

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7. Comprehensive Worked Numerical Example: Stefan Tube Evaporator Rating

Problem Statement

A chemical plant storage tank vent incorporates a vertical Stefan tube apparatus to quantify solvent evaporation losses. Liquid benzene ($A$, $\text{C}_6\text{H}_6$, $M_A = 78.11\text{ g/mol}$) is maintained at the bottom of a vertical tube at a constant temperature of $T = 298.15\text{ K}$ ($25.0^\circ\text{C}$) and an ambient total pressure of $P = 1.00\text{ atm} = 101.325\text{ kPa}$.

The distance from the liquid benzene meniscus to the open top of the tube is fixed at $z = 0.150\text{ m}$ ($15.0\text{ cm}$). Dry air ($B$, insoluble in liquid benzene) flows continuously across the top of the tube at high velocity such that the partial pressure of benzene at the tube exit is effectively zero.

Given Physical Data at $25.0^\circ\text{C}$:

• Vapor pressure of benzene: $P_A^* = 100.0\text{ mmHg} = 13.332\text{ kPa} = 0.13158\text{ atm}$.
• Binary gas diffusivity of benzene in air: $D_{AB} = 8.80 \times 10^{-6}\text{ m}^2/\text{s}$.
• Universal gas constant: $R = 8.31446\text{ J/(mol}\cdot\text{K)} = 8.31446\text{ Pa}\cdot\text{m}^3/(\text{mol}\cdot\text{K})$.

Calculate:

1. The partial pressures of air at the liquid interface ($p_{B1}$) and at the tube exit ($p_{B2}$).
2. The log-mean partial pressure of air ($p_{B,lm}$) and the Stefan drift enhancement factor ($P / p_{B,lm}$).
3. The steady-state evaporative molar flux of benzene ($N_A$) in $\text{mol/(m}^2\cdot\text{s)}$.
4. The molar flux that would occur if equimolar counterdiffusion prevailed ($N_{A,\text{EMCD}}$), and the percentage increase attributable to bulk drift.
5. The areal mass loss rate of benzene in $\text{kg/(m}^2\cdot\text{hr)}$.

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Step 1: Boundary Partial Pressures

At the liquid meniscus ($z = 0$):

• Benzene is in thermodynamic phase equilibrium with its pure liquid: $$p_{A1} = P_A^* = 13,332.2\text{ Pa} = 0.13158\text{ atm}$$
• Air partial pressure is determined by Dalton's Law: $$p_{B1} = P - p_{A1} = 101,325\text{ Pa} - 13,332.2\text{ Pa} = \mathbf{87,992.8\text{ Pa}} = 0.86842\text{ atm}$$

At the open tube exit ($z = 0.150\text{ m}$):

• Air sweep strips benzene completely: $$p_{A2} = 0.0\text{ Pa}$$
• Air partial pressure equals total pressure: $$p_{B2} = P - p_{A2} = \mathbf{101,325.0\text{ Pa}} = 1.0000\text{ atm}$$

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Step 2: Log-Mean Partial Pressure and Stefan Drift Factor

Evaluating the logarithmic mean of the stagnant carrier gas:

$$p_{B,lm} = \frac{p_{B2} - p_{B1}}{\ln(p_{B2} / p_{B1})} = \frac{101,325.0 - 87,992.8}{\ln(101,325.0 / 87,992.8)}$$ $$\frac{p_{B2}}{p_{B1}} = \frac{101,325.0}{87,992.8} = 1.151515$$ $$\ln(1.151515) = 0.141072$$ $$p_{B,lm} = \frac{13,332.2\text{ Pa}}{0.141072} = \mathbf{94,506.5\text{ Pa}} = 0.93271\text{ atm}$$

The Stefan drift enhancement factor is:

$$\frac{P}{p_{B,lm}} = \frac{101,325.0\text{ Pa}}{94,506.5\text{ Pa}} = \mathbf{1.07215} \approx \mathbf{1.072}$$

Bulk convective flow enhances diffusion by $7.22\%$ above pure molecular diffusion.

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Step 3: Evaporative Molar Flux Calculation

Using the stagnant diffusion formula:

$$N_A = \frac{D_{AB} P}{R T z p_{B,lm}} (p_{A1} - p_{A2})$$

Evaluate the product $R T z$:

$$R T z = (8.31446\text{ Pa}\cdot\text{m}^3/(\text{mol}\cdot\text{K})) \times (298.15\text{ K}) \times (0.150\text{ m})$$ $$R T z = 2,478.96 \times 0.150 = 371.843\text{ Pa}\cdot\text{m}^4/\text{mol}$$

Substitute all values in consistent SI units:

$$N_A = \frac{(8.80 \times 10^{-6}\text{ m}^2/\text{s}) \times (101,325\text{ Pa})}{(371.843\text{ Pa}\cdot\text{m}^4/\text{mol}) \times (94,506.5\text{ Pa})} \times (13,332.2\text{ Pa} - 0)$$ $$N_A = \frac{0.89166}{(371.843) \times (94,506.5)} \times 13,332.2 = \frac{0.89166 \times 13,332.2}{35,141,600}$$ $$N_A = \frac{11,887.8}{35,141,600} = \mathbf{3.3828 \times 10^{-4}\text{ mol/(m}^2\cdot\text{s)}} = \mathbf{0.3383\text{ mmol/(m}^2\cdot\text{s)}}$$

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Step 4: Comparison to Equimolar Counterdiffusion (EMCD)

If equimolar counterdiffusion were incorrectly assumed:

$$N_{A,\text{EMCD}} = \frac{D_{AB}}{R T z} (p_{A1} - p_{A2}) = \frac{8.80 \times 10^{-6}}{371.843} \times 13,332.2 = \mathbf{3.1552 \times 10^{-4}\text{ mol/(m}^2\cdot\text{s)}}$$

Ratio verification:

$$\frac{N_A}{N_{A,\text{EMCD}}} = \frac{3.3828 \times 10^{-4}}{3.1552 \times 10^{-4}} = \mathbf{1.0721} = \frac{P}{p_{B,lm}}$$

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Step 5: Areal Mass Loss Rate of Benzene

Convert molar flux to hourly mass emission rate per unit area:

$$\dot{m}_A'' = N_A \cdot M_A = (3.3828 \times 10^{-4}\text{ mol/(m}^2\cdot\text{s)}) \times (78.11\text{ g/mol}) = 0.026423\text{ g/(m}^2\cdot\text{s)}$$ $$\dot{m}_A'' = 0.026423\text{ g/(m}^2\cdot\text{s)} \times \frac{3,600\text{ s}}{1\text{ hr}} \times \frac{1\text{ kg}}{1,000\text{ g}} = \mathbf{0.09512\text{ kg/(m}^2\cdot\text{hr)}}$$

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8. Critical NCEES PE Exam Traps & Pitfalls

Trap 1: Omitting the $P / p_{B,lm}$ Factor in Evaporator and Scrubber Sizing
Whenever an exam question describes evaporation of a liquid into a gas blanket or absorption of a solute from an inert carrier (such as air, $\text{N}_2$, or methane), the non-volatile/insoluble gas is stagnant ($N_B = 0$). Neglecting the log-mean factor $P / p_{B,lm}$ underpredicts mass flux by $5\%$ to $50\%$ depending on solute concentration. Always check whether $N_B = -N_A$ (EMCD) or $N_B = 0$ (stagnant $B$).

Trap 2: Incorrect Temperature and Pressure Scaling of Diffusivities
Candidates frequently confuse the scaling of gas versus liquid diffusivities:

• Gas Diffusivity: $D_{AB} \propto T^{1.75} / P$. Doubling the absolute pressure halves the gas diffusivity ($D_{AB} \propto 1/P$).
• Liquid Diffusivity: $D_{AB} \propto T / \mu_B$. Moderately increasing total pressure from $1\text{ bar}$ to $10\text{ bar}$ has zero effect on liquid diffusivity because liquids are incompressible, but temperature changes dramatically alter diffusivity via solvent viscosity $\mu(T)$.

Trap 3: Confusing Diffusive Flux ($J_A^*$) with Total Absolute Flux ($N_A$)
$J_A^*$ is defined relative to the mixture's moving molar-average velocity ($v^*$), whereas $N_A$ is measured relative to stationary apparatus walls. On the PE exam, equipment mass balances always require the absolute flux $N_A$. The two are identical only in Equimolar Counterdiffusion ($N_A + N_B = 0$) or in infinitely dilute solutions ($x_A \to 0$).